# Three Vertices of Parallelogram Abcd Taken in Order Are A(3, 6), B(5, 10) and C(3, 2) the Coordinate of the Fourth Vertex D Length of Diagonal Bd Equation of the Side Ad of the Parallelogram Abcd - Mathematics

Three vertices of parallelogram ABCD taken in order are A(3, 6), B(5, 10) and C(3, 2)

1) the coordinate of the fourth vertex D

2) length of diagonal BD

3) equation of the side AD of the parallelogram ABCD

#### Solution

Three vertices of a parallelogram taken in order are A(3, 6), B(5, 10) and C(3, 2)

1) We need to find the coordinates of D.

We know that the diagonals of a parallelogram bisect each other.

Let x, y be the coordinates of D.

∴ Mid-point of diagonal AC = ((3+3)/2, (6+2)/2) = (3,4)

And midpoint of diagonal BD = ((5+x)/2, (10 + y)/2)

Thus we have

(5 + x)/2 = 3 and (10 + y)/2 = 4

=> 5  + x = 6  and 10 + y = 8

=> x = 1 and  y = -2

:. D = (1,-2)

2) Lenght of diagonal BD = sqrt((1 - 5)^2 + (-2-10)^2)

= sqrt((-4)^2 + (-12)^2)

= sqrt(16 + 144)

= sqrt(160)

= 4sqrt10

3) A(3,6) = (x_1. y_1) and B(5,10) = (x_2, y_2)

Slope of line AB = m(y_2 - y_1)/(x_2 - x_1) = (10 - 6)/(5-3) = 4/2 = 2

∴ Equation of line AB is given by

y - y_1 = m(x - x_1)

=> y - 6 = 2(x - 3)

=> y - 6 = 2x - 6

=> 2x - y = 0

=> 2x = y

Concept: Simple Applications of All Co-ordinate Geometry.
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